Lectures on Symplectic Geometry by Ana Cannas da Silva

Lectures on Symplectic Geometry - Table of Contents

INTRODUCTION PART I — SYMPLECTIC MANIFOLDS 1. Symplectic Forms 2. Symplectic Form on the Cotangent Bundle PART II — SYMPLECTOMORPHISMS 3. Lagrangian Submanifolds 4. Generating Functions 5. Recurrence PART III — LOCAL FORMS 6. Preparation for the Local Theory 7. Moser Theorems 8. Darboux–Moser–Weinstein Theory 9. Weinstein Tubular Neighborhood Theorem PART IV — CONTACT MANIFOLDS 10. Contact Forms 11. Contact Dynamics PART V — COMPATIBLE ALMOST COMPLEX STRUCTURES 12. Almost Complex Structures 13. Compatible Triples 14. Dolbeault Theory PART VI — KÄHLER MANIFOLDS 15. Complex Manifolds 16. Kähler Forms 17. Compact Kähler Manifolds PART VII — HAMILTONIAN MECHANICS 18. Hamiltonian Vector Fields 19. Variational Principles 20. Legendre Transform PART VIII — MOMENT MAPS 21. Actions 22. Hamiltonian Actions PART IX — SYMPLECTIC REDUCTION 23. The Marsden–Weinstein–Meyer Theorem 24. Reduction PART X — MOMENT MAPS REVISITED 25. Moment Map in Gauge Theory 26. Existence and Uniqueness of Moment Maps 27. Convexity PART XI — SYMPLECTIC TORIC MANIFOLDS 28. Classification of Symplectic Toric Manifolds 29. Delzant Construction 30. Duistermaat–Heckman Theorems

What You Will Learn in Lectures on Symplectic Geometry

"Lectures on Symplectic Geometry" by "Ana Cannas da Silva" is a clear and well-structured introduction to "symplectic geometry", written for graduate students and mathematically mature readers. The book grows out of lecture notes and is designed to bridge the gap between basic "differential geometry" and modern research topics. It assumes familiarity with manifolds and calculus on forms, making it ideal for readers moving beyond introductory geometry. The text develops the subject step by step, starting with symplectic forms and symplectic manifolds, and moving toward deeper ideas such as "Hamiltonian mechanics", moment maps, and symplectic reduction. Key results like Darboux’s theorem are presented with strong geometric intuition, helping readers understand not just the statements, but why they matter. The book also explores links to "Lie groups", contact geometry, and Kähler manifolds, showing how symplectic geometry fits into a broader mathematical framework. What makes this book especially valuable is its balance between rigor and accessibility. The writing is concise and human-friendly, with examples and exercises that reinforce understanding. Overall, it is an excellent foundation for anyone planning further study or research in "modern geometry" or mathematically oriented physics.

Book Details & Specifications

Title: Lectures on Symplectic Geometry by Ana Cannas da Silva
Publisher: Springer
Year: 2006
Pages: 225
Type: PDF
Language: English
ISBN-10 #: 3540421955
ISBN-13 #: 978-3540421955
License: External Educational Resource
Amazon: Amazon

About the Author: Ana Cannas da Silva

The author Ana Cannas da Silva is a "Portuguese mathematician" best known for her contributions to "symplectic geometry" and geometric topology. She earned her PhD from "MIT" under Victor Guillemin and has held academic positions at leading institutions, including Princeton and UC Berkeley. She is currently a professor at "ETH Zurich", where she teaches and researches modern geometry. Her book "Lectures on Symplectic Geometry" is widely used by graduate students and researchers, valued for its clear exposition and strong connection between "differential geometry", topology, and classical mechanics.

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